Rotate object having animations
You could try parenting the character rig to something else, maybe an empty or a non-printing object, that is centered right at the base of the rig. Then rotate that parent object, and everything should be good.

Actually, rigs often have a base object included. Check to see if your character's rig already has a base or handle that everything else is parented to. Then instead or rotating the whole character, just rotate the base. You should get better result.